Geometric Lorenz flows with historic behavior
We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.
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arXiv subjects
Ming-Chia Li
Publications and source records attributed to Ming-Chia Li.
Covering relations for coupled map networks
Following [6,12], we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks is obtained. The result is quite general, particularly no assumptions on hyperbolicity of a local dynamics or linearity of coupling are made.
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Coexistence of invariant sets with and without SRB measures in Hénon family
Let $\{f_{a,b}\}$ be the (original) Hénon family. In this paper, we show that, for any $b$ near $0$, there exists a closed interval $J_b$ which contains a dense subset $J'$ such that, for any $a\in J'$, $f_{a,b}$ has a quadratic homoclinic tangency associated with a saddle fixed point of $f_{a,b}$ which unfolds generically with respect to the one-parameter family $\{f_{a,b}\}_{a\in J_b}$. By applying this result, we prove that $J_b$ contains a residual subset $A_b^{(2)}$ such that, for any $a\in A_b^{(2)}$, $f_{a,b}$ admits the Newhouse phenomenon. Moreover, the interval $J_b$ contains a dense subset $\tilde A_b$ such that, for any $a\in \tilde A_b$, $f_{a,b}$ has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously. Dedicated to the memory of Floris Takens (Nov. 12, 1940 - Jun. 20, 2010).
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Bifurcation of the ACT map
In this paper, we study the Arneodo-Coullet-Tresser map $ F(x,y,z)=(ax-b(y-z), bx+a(y-z), cx-dx^k+e z)$ where $a,b,c,d,e$ are real with $bd\neq 0$ and $k>1$ is an integer. We obtain stability regions for fixed points of $F$ and symmetric period-2 points while $c$ and $e$ vary as parameters. Varying $a$ and $e$ as parameters, we show that there is a hyperbolic invariant set on which $F$ is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of $F$ while $c$ and $d$ vary as parameters and $F$ is near an anti-integrable limit. Some numerical results indicates $F$ has Hopf bifurcation, strange attractors, and nested structure of invariant tori.
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A refinement of Sharkovskii's theorem on orbit types characterized by two parameters
The so-called type problem or forcing problem is considered as a way to generalize Sharkovskii's theorem. In this paper, by focusing on certain types of orbits, we obtain a solution of the type problem, which gives a refinement of Sharkovskii's theorem on orbit types characterized by two parameters.
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